Infinite BS & Orders of Infinity

Something that bothers me is the blind, stupefied passivity with which people, and even academic institutions, have accepted blather about “orders of infinity” and even “crunched dimensions” - contradictions in terms axiomatically not to be posited. It is as if squaring the circle were still considered possible, and worse - but I have noticed on the Net claimants to status as bordering on succeeding on doing just that. It seems to relate to insanity, crossed mental circuits, some kind of semi-epileptic, amphetamine-like monomania. Infinite means just that – no limits, of extent beyond anything else, and an infinite number of elephants and an infinite number of ants are the same size, just as a ton of lead weighs exactly as much as a ton of feathers! And a dimension extends infinitely - you simply can’t hope to put one in your pocket, no matter what you might have encountered in the comics.
But as the concepts of zero and one have no actual, unequivocal analogue in the perceptible world (nature “abhorring” a vacuum, and everything interacting with everything and lacking pure boundaries or the hard edges we assume as a convenience), and fuzzy logic has given us great screen-picture resolution, among other things, I suppose there’ll be no stopping the blather.
It all reminds me of Yale linguists Benjamin Whorf and Edward Sapir claiming Hopi language has no word for time, and that Hopi time-sense is circular. I expect that the Sapir–Whorf hypothesis, that the structure of a language tends to condition the ways in which a speaker of that language thinks (and that the structures of different languages lead the speakers of those languages to view the world in different ways) is correct, but it wasn’t even either of them who originated the idea, and apparently they didn’t talk to any Hopis about it, or, if they did, the Hopi they met must have pulled their leg the way the Dogon did with some anthropologist, claiming they came from the dark star Sirius B, a prank long ago discredited but still reflected in Encyclopedia Britannica’s entry on the Dogon. The Hopi, and others, have done this kind of thing to get a laugh out of meddlesome interlopers not really willing to spend much time - I knew a young white man who some Tewa from the Hopi First Mesa had also known… he’d bragged to me about his initiation into the Chakwaina Society. The Chakwaina Katchina was modeled on the first Black man Pueblo people encountered (so he’d been made a kind of honorary nigger).
It can be so good to feel important, have something interesting to say, to pose as an achiever… I worry about falling into the same pit here.

At any rate, there’s a curious notion that one ‘infinity’ can be bigger than another (My Infinity’s bigger than your infinity, always has been and always will be). Perhaps it comes from a misguided notion that infinity is a real, instead of imaginary, number.

Recently, a now ex friend told me that I shouldn’t challenge notions I don’t understand; that some people know more than me, especially about math. Now, I’m fine to admit that some people know more about the commodities market than I do (I know scarce little about it), but somehow I think that doesn’t apply to commodities futures, or ability to think one’s way out of a paper box. I’ve learned how to think. Most people, including most academics, haven’t.

Tired from 45 days of hosting a tiresome guest (that now ex-friend), it took me a few hours to think out a word problem attributed to Frank Morgan (“The Math Chat Book”, Math Association of America 2000), which I found in John D. Barrow’s “The Infinity Book (Vintage, Random House, Great Britain, 2005). “Three guys to into a hotel, each with $10 in his pocket. They book one room at $30 a night. A short while later a fax from headquarters directs the hotel to charge $25 a night. So the receptionist gives the bellhop $5 to take to the three guys sharing a room. Since the bellhop never got a tip from them and because he can’t split $5 three ways, he decides to pocket $2 and give them each one dollar back. So each of the three guys has not spent $9 and the bellhop has $2, for a total of $29 Where’s the extra dollar?”

It’s misdirection, misapplied association, and basically the same thing used in the next chapter to “prove” (a la Georg Cantor) that some infinities are “bigger” (or “stronger”) than others. Now, Barrow does quote Galileo: “we cannot speak of infinite quantities as being the one greater or less than or equal to another”, and narrate some about Cantor’s problems with mental illness (not uncommon among those who’ve wrestled with problems resultant from assuming there can be “orders of infinity”). Cantor’s sleight of hand involves distinguishing “countable” and “uncountable” infinities, then proclaiming that, since a one-to-one correspondence of irrational numbers and rational numbers cannot be made (and there are even problems ordering irrational, and most certainly imaginary, numbers), the sets cannot be equal (the set of irrational numbers is “infinitely bigger than the natural numbers or the fractions.”).
One problem with this had already been addressed in Barrow’s book: in the 14th century, Albert of Saxony showed that an infinite amount of something bigger does not represent more than an infinite amount of something smaller. “Take an infinitely long beam of wood, with a square cross-section of size 1 unit by 1 unit. Now saw it up into cubes of equal size. You will have an infinite number of these cubes which you can now use as building blocks.” With these blocks, you could “fill the whole of an infinite three-dimensional space!” The infinite has strange properties, including not becoming smaller when something is subtracted from it… and not needing a one-to-one correspondence with another infinity.
One of Barrow’s chapters is titled, “Infinity Comes in Three Flavours”. They are mathematical infinities, physical infinities (!) and absolute infinity (!!!). Interestingly, in George Gamow’s “One Two Three… Infinity” (© 1947, 1961), there are reputed to be only three orders of infinity: the number of all integers, the number of all geometrical points, and the number of all curves. These are supposedly “enough to count anything we can think of” – except imaginary numbers. And, to give them credit, mathematicians have certainly shown some imagination. But, then again, much of modern math simply doesn’t apply to the world we can experience.

Not yet half-way through Barrow’s book, I was impressed with his discussion of “string theory”. He describes Roger Penrose’s proposal that there’s “a principle of ‘cosmic censorship’ in nature, so that all singularities, or physical infinities, where the laws of Nature break down, are hidden from the outside universe by horizon surfaces.” I’ll leave it to the reader to interpret that, or not. After a bit, he goes on, “Up until 1974, black holes were believed to be inescapable matter traps. Once you passed in through the horizon there was no escape. Then Stephen Hawking predicted that black holes should not be completely black. Their strong gravitational fields will gradually produce pairs of particles close to the horizon at the expense of the mass and energy of the black hole. Gradually the mass of the hole will evaporate away.” Then, “Physical infinities will be nuanced by the laws of Nature.” Barrow claims that string theorists (well, “particle physicists”) don’t believe in physical infinities (which is fine by me). “Since the 1980s, string theory has shown how these infinities can be avoided by changing our conception of what the most elementary pieces of the Universe look like.” Well, of course, they don’t “look” like anything – you can’t see them. But it can be postulated that they have shape… well, sort of, anyway. Instead of things imploding into a crunch of infinite density, a ‘bounce’ at finite density and temperature results “from a previous state of contraction, or by a non-expanding state that suddenly bursts into motion.”
Reminding me of word games like Noam Chomsky trying to use words to explain words (it can’t be done, anymore than my infinity can be truly shown to be larger than yours, see Gödel's Theorem). But Chomsky the political critic makes some interesting points, including that, if the Nuremberg laws were applied, then “every post-war American president would have been hanged.” At a news conference in 1977, President Jimmy Carter said about the Vietnam War, “We have no need to apologize or castigate ourselves or assume the status of culpability. We do not owe a debt. Our intentions were to defend the freedoms of the South Vietnamese, and the destruction was mutual.” Never-mind that the Vietnamese saw themselves as defending their homes. Word-tricksters have been at it for thousands of years, calling the opposition, pre-emptively, what they know they are themselves, or God “That than which no greater can be conceived”, or telling you that you have no right to criticize “authority” as you simply do not know enough.
Some try to justify "orders of infinity" through a trick of mis-direction whereinwhich it's shown that some sets have no one-to-one relationship with others, and therefor must be larger. This is spurious. The concepts of infinity and one-to-one relationships have no direct, or logical, connection. To compare one infinite series to another establishes nothing, but perhaps the fact that so many have believed that it does establish establish something, itself establishes something. People will believe anything they find reason to want to believe.